Let A = {1, 2, 3, 4, 5}. Let R be a relation on A defined by xRy if and only if 4x ≤ 5y. Let m be th

Q: Wait, why do we include $(5,4)$ in the original set $R$?

The condition is $4x \le 5y$. For the pair $(5,4)$, we have $x=5$ and $y=4$. Substituting these, $4(5) = 20$ and $5(4) = 20$. Since $20 \le 20$ is true, $(5,4)$ is definitely in $R$.

Q: Do we need to add $(1,1), (2,2), etc.$ to make it symmetric?

No! A relation is symmetric if $(x, y) \in R \implies (y, x) \in R$. For diagonal elements like $(1,1)$, the pair $(y, x)$ is just $(1,1)$, which is already in $R$. So, they are already symmetric.

Q: Is there a faster way to count $m$?

Yes, you can count the points in the grid. For each $x$, count how many $y$ values satisfy $y \ge 0.8x$. Summing these counts is the most reliable method. For $x=1$, all $5$ work. For $x=2$, $y \ge 1.6$, so $4$ work. And so on.

Question

Let $A = \{1, 2, 3, 4, 5\}$. Let $R$ be a relation on $A$ defined by $xRy$ if and only if $4x \le 5y$. Let $m$ be the number of elements in $R$ and $n$ be the minimum number of elements from $A 	imes A$ that are required to be added to $R$ to make it a symmetric relation. Then $m + n$ is equal to :

Options:

(A) (1) 25

(B) (2) 24

(C) (3) 26

(D) (4) 23

Sigma Percentile · Accepted Answer

### The Geometry of Relations

Welcome, fellow traveler on the path to JEE mastery! Today, we are going to explore the elegant world of relations.

Often, students see a relation like $xRy \iff 4x \le 5y$ and feel intimidated by the abstract notation. But let's strip away the fear and look at the geometry.

Imagine a $5 	imes 5$ grid where the $x$-axis and $y$-axis both represent the set $A = \{1, 2, 3, 4, 5\}$. Each point $(x, y)$ on this grid is a potential member of our relation $R$.

The condition $4x \le 5y$ is not just an inequality; it is a boundary. If we rearrange it, we get:

$$y \ge \frac{4}{5}x$$

This means that for any $x$, we are looking for all $y$ values that lie on or above the line $y = \frac{4}{5}x$. This is our territory.

### Mapping the Territory: Finding $m$

Let's systematically map out our points. For $x=1$, the condition is $y \ge 0.8$. Since $y$ must be an integer in $A$, all $y \in \{1, 2, 3, 4, 5\}$ satisfy this. That gives us $5$ points: $(1,1), (1,2), (1,3), (1,4), (1,5)$.

Moving to $x=2$, we need $y \ge 1.6$. The valid integers are $2, 3, 4, 5$. That is $4$ points: $(2,2), (2,3), (2,4), (2,5)$.

For $x=3$, we need $y \ge 2.4$. The valid integers are $3, 4, 5$. That is $3$ points: $(3,3), (3,4), (3,5)$.

For $x=4$, we need $y \ge 3.2$. The valid integers are $4, 5$. That is $2$ points: $(4,4), (4,5)$.

Finally, for $x=5$, we need $y \ge 4.0$. The valid integers are $4, 5$. That is $2$ points: $(5,4), (5,5)$.

Summing these up, $m = 5 + 4 + 3 + 2 + 2 = 16$. We have successfully identified the **$16$ elements** of our relation $R$.

### The Mirror Test: Symmetry

A relation is symmetric if, for every pair $(x, y) \in R$, the pair $(y, x)$ is also in $R$. Geometrically, this is a reflection across the line $y = x$.

If our relation were symmetric, our set of points would be perfectly balanced on both sides of this diagonal. But is it? Let's check.

We have $(1,2)$ in $R$, but is $(2,1)$ in $R$? Checking the condition:

$$4(2) \le 5(1) \Rightarrow 8 \le 5$$

This is false. So, $(2,1)$ is missing! We need to add these missing counterparts.

### Identifying Missing Pairs

Let's list the pairs $(x, y) \in R$ and check if $(y, x) \in R$:

*   $(1,2) \in R$, but $(2,1) 
otin R$. (Need to add $(2,1)$)
*   $(1,3) \in R$, but $(3,1) 
otin R$. (Need to add $(3,1)$)
*   $(1,4) \in R$, but $(4,1) 
otin R$. (Need to add $(4,1)$)
*   $(1,5) \in R$, but $(5,1) 
otin R$. (Need to add $(5,1)$)
*   $(2,3) \in R$, but $(3,2) 
otin R$. (Need to add $(3,2)$)
*   $(2,4) \in R$, but $(4,2) 
otin R$. (Need to add $(4,2)$)
*   $(2,5) \in R$, but $(5,2) 
otin R$. (Need to add $(5,2)$)
*   $(3,4) \in R$, but $(4,3) 
otin R$. (Need to add $(4,3)$)
*   $(3,5) \in R$, but $(5,3) 
otin R$. (Need to add $(5,3)$)
*   $(4,5) \in R$, and $(5,4) \in R$. (Already symmetric!)

### The Final Tally

Counting the missing pairs, we have $4 + 3 + 2 = 9$ pairs that must be added to achieve symmetry. Thus, $n = 9$.

The total number of elements required to make the relation symmetric is:

$$m + n = 16 + 9 = 25$$

Isn't it beautiful? By visualizing the relation as a set of points on a grid, we transformed a daunting algebraic problem into a simple counting exercise. Keep this geometric intuition in your toolkit, and no relation problem will ever stand in your way again!

Select Answer:

Visualized Solution

Understanding the Set and Relation

Defining the Relation $R$

Finding Pairs for $x = 1$

Finding Pairs for $x = 2$

Finding Pairs for $x = 3$

Finding Pairs for $x = 4$ and $x = 5$

Calculating $m$

Understanding Symmetry

Adding Symmetric Counterparts

Completing the Symmetric Closure

Calculating $n$

Final Calculation: $m + n$

The Sigma Insight: Types of Relations

The Geometry of Relations

Mapping the Territory

The Mirror Test

Identifying Missing Pairs

The Final Tally

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